Fascism is a reaction to a capitalist crisis, Nazis weren't socialists by any definition except for using populist agendas like sharing of profits and welfare to rally their base.
Nazis never nationalised any industry and rather used corporatism to take hold of them while those were still private entities, same in Italy, industry and fascist regimes walked hand-in-hand.
Saying those are equivalent to socialism just due to populist rhetoric is misleading and historically wrong...
Saying they are free market capitalists is misleading and historically wrong.
This complaint is also like saying communists weren’t authentic because the government withheld power from the Soviet’s and used the Soviet narrative to gain popular support.
We compare ideologies on a political spectrum. It’s harder to compare implementations because they all look like corruption. Hitler is one person in a political movement. He did write the platform I linked to though.
They did actually implement many of these things, such as universal healthcare (for their race).